Solve 3x^2-15x^2+75x+1000

$$3 x ^ { 2 } - 15 x ^ { 2 } + 75 x + 1000$$

Factor

$$-12\left(x-\left(-\frac{5\sqrt{2145}}{24}+\frac{25}{8}\right)\right)\left(x-\left(\frac{5\sqrt{2145}}{24}+\frac{25}{8}\right)\right)$$

Steps Using the Quadratic Formula

Combine 3x^{2} and -15x^{2} to get -12x^{2}.

$$factor(-12x^{2}+75x+1000)$$

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-12x^{2}+75x+1000=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-75±\sqrt{75^{2}-4\left(-12\right)\times 1000}}{2\left(-12\right)}$$

Square 75.

$$x=\frac{-75±\sqrt{5625-4\left(-12\right)\times 1000}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-75±\sqrt{5625+48\times 1000}}{2\left(-12\right)}$$

Multiply 48 times 1000.

$$x=\frac{-75±\sqrt{5625+48000}}{2\left(-12\right)}$$

Add 5625 to 48000.

$$x=\frac{-75±\sqrt{53625}}{2\left(-12\right)}$$

Take the square root of 53625.

$$x=\frac{-75±5\sqrt{2145}}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{-75±5\sqrt{2145}}{-24}$$

Now solve the equation x=\frac{-75±5\sqrt{2145}}{-24} when ± is plus. Add -75 to 5\sqrt{2145}.

$$x=\frac{5\sqrt{2145}-75}{-24}$$

Divide -75+5\sqrt{2145} by -24.

$$x=-\frac{5\sqrt{2145}}{24}+\frac{25}{8}$$

Now solve the equation x=\frac{-75±5\sqrt{2145}}{-24} when ± is minus. Subtract 5\sqrt{2145} from -75.

$$x=\frac{-5\sqrt{2145}-75}{-24}$$

Divide -75-5\sqrt{2145} by -24.

$$x=\frac{5\sqrt{2145}}{24}+\frac{25}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{25}{8}-\frac{5\sqrt{2145}}{24} for x_{1} and \frac{25}{8}+\frac{5\sqrt{2145}}{24} for x_{2}.

$$-12x^{2}+75x+1000=-12\left(x-\left(-\frac{5\sqrt{2145}}{24}+\frac{25}{8}\right)\right)\left(x-\left(\frac{5\sqrt{2145}}{24}+\frac{25}{8}\right)\right)$$

Evaluate

$$1000+75x-12x^{2}$$

Solution Steps

Combine 3x^{2} and -15x^{2} to get -12x^{2}.

$$-12x^{2}+75x+1000$$